Math Problem of the Day

Let $q(i)$ be the $i^\text{th}$ smallest integer that can be written as $F_x + 2^y$, where $x, y \in \mathbb{Z}$, $x > 0$, $y \ge 0$, and $F_x$ is the $x^\text{th}$ Fibonacci number. There are $105$ kids, numbered $1$ through $105$, such that the $i^\text{th}$ kid lives at coordinate $q(i)$ on a number line. We want to place a dinosaur at the coordinate $T$ such that the total distance that each kid has to travel to reach the dinosaur is minimized. Find $T$.